Global eigenmodes of thin liquid sheets by means of Volume-of-Fluid simulations

Abstract

The unsteady dynamics of planar liquid sheet flows, interacting with unconfined gaseous environments located on both sides of the liquid phase, is numerically investigated by means of the Volume-of-Fluid (VOF) technique for supercritical regimes. The global behavior of the non-parallel flow is analyzed by perturbing the initial steady configuration by means of a Gaussian bump in the transverse velocity component of relatively small amplitude, thereby exciting sinuous modes. To gain more physical insights into the fluid system, a theoretical linear one-dimensional model is also developed. A physical interpretation of this model relates the sheet dynamics to transverse vibrations of tensional string forced by terms containing the lateral velocity and subjected to a total damping coefficient, which can assume negative values. The VOF simulation satisfactorily confirms that the velocity impulse perturbation splits into two wave fronts traveling downstream with the theoretical wave velocities. A good agreement is found in comparing the crossing times over the entire domain length of such waves with the almost constant spacing between the frequencies of the eigenvalue spectrum. Surface tension plays a stabilizing role, and for relatively high values of density ratio rρ of gaseous-to-liquid phases, the sheet becomes unstable. It is argued that the distribution of transverse velocity component of the gaseous phase represents the forcing term, which leads the system toward the instability when, for relatively high rρ, the total damping becomes negative. An analogy seems to exist between the global unstable behavior exhibited by the liquid sheet as rρ increases and the shear-induced global instability found by Tammisola et al. [Surface tension-induced global instability of planar jets and wakes,” J. Fluid Mech. 713, 632–658 (2012)] in the presence of surface tension. However, for the gravitational sheet, the surface tension is stabilizing.

Natural frequency discontinuity of vertical liquid sheet flows at transcritical threshold

“…(2017) and Della Pia et al. (2020, 2021), the supercritical regime is characterized by the presence of two branches exhibiting an almost constant spacing between the imaginary part of the eigenvalues (frequency), which is directly associated with the crossing time of slow (upper branch, ) and fast (lower branch, ) travelling waves (i.e. with velocity relative to that of the base flow , respectively) featuring the solution of and .…”

“…(2017) (see also (34) in Della Pia et al. 2020). Based on this consideration, we set up the linear stability analysis as described in § 2.1, to predict the natural frequency of the finite curtain flow and provide comparisons with the experimental findings reported in § 4.…”

“…Both the natural and forced responses of a gravitational liquid sheet, interacting with an unconfined gaseous ambient, are analysed by means of a linear inviscid one-dimensional model, described in detail in previous contributions (Della Pia et al. 2020, 2021). Despite the simplifying assumptions, the predictions of this model were found to be valuable compared with two-phase numerical simulations carried out with a volume-of-fluid (VOF) code (Popinet 2003, 2009).…”

“…In supercritical conditions, the natural frequency of the system was predicted with a relative spread of (Della Pia et al. 2020). In the subcritical regime, instead, an agreement within was found between the natural frequency arising from the model and the resonance frequency of the VOF simulated sheet subjected to a time continuous sinusoidal transverse velocity forcing applied at the inlet section (Della Pia et al.…”

“…1997; Lin 2003; Barlow, Weinstein & Helenbrook 2012). Nowadays, further insights can be provided by direct numerical simulation of the relevant two-phase flow field (Della Pia, Chiatto & de Luca 2020), which enables the solution of the curtain motion and of the surrounding gaseous ambient to be predicted simultaneously. The free and forced dynamic responses of such a flow system have been investigated for the cases of sheet interacting with a one-sided air cushion (nappe configuration, Binnie 1974; Sato et al.…”